Evaluate the logarithm. Round your answer to the nearest thousandth. log_(3)((1)/(45))~~

Problem Understanding: Understand the problem. We need to evaluate the logarithm of the fraction 1/45 with base 3 and round the result to the nearest thousandth.Simplifying the Expression: Use the logarithm properties to simplify the expression. We can use the property that log(a/b) = log(a) - log(b) to rewrite the given logarithm as: log_(3)(1) - log_(3)(45)Evaluating log(1): Evaluate log_(3)(1). The logarithm of 1 to any base is 0 because any number to the power of 0 is 1. So, log_(3)(1) = 0.Factoring 45: Factor 45 to express it as a product of powers of 3. 45 can be factored into 3 * 15, and 15 can be further factored into 3 * 5. So, 45 = 3^2 * 5.Separating the Factors: Use the logarithm properties to separate the factors. We can use the property that log(ab) = log(a) + log(b) to rewrite log_(3)(45) as: log_(3)(3^2) + log_(3)(5)Evaluating log(3^2): Evaluate log_(3)(3^2). Since the base of the logarithm and the base of the exponent are the same, log_(3)(3^2) = 2.Evaluating log(5): Evaluate log_(3)(5). Since 5 is not a power of 3, we cannot simplify this logarithm further without a calculator. We will need to use a calculator to find the value of log_(3)(5).Calculating log(3^2): Calculate log_(3)(5) using a calculator. Using a calculator, we find that log_(3)(5) ≈ 1.465 (rounded to the nearest thousandth).Combining the Results: Combine the results to find the final value of the original logarithm. Now we combine the results from steps 3, 6, and 8: log_(3)((1)/(45)) = log_(3)(1) - (log_(3)(3^2) + log_(3)(5)) = 0 - (2 + 1.465) = -2 - 1.465 = -3.465Rounding the Final Answer: Round the final answer to the nearest thousandth. The final answer is already rounded to the nearest thousandth, so the final answer is -3.465.

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Evaluate the logarithm.
Round your answer to the nearest thousandth.

log_(3)((1)/(45))~~

Evaluate the logarithm. $\newline$ Round your answer to the nearest thousandth. $\newline$ $\log _{3}\left(\frac{1}{45}\right) \approx$

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Algebra 1

Compare linear, exponential, and quadratic growth

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Q. Evaluate the logarithm.

\newline

Round your answer to the nearest thousandth.

\newline

\log _{3}\left(\frac{1}{45}\right) \approx

Problem Understanding: Understand the problem. $\newline$ We need to evaluate the logarithm of the fraction $\frac{1}{45}$ with base $3$ and round the result to the nearest thousandth.
Simplifying the Expression: Use the logarithm properties to simplify the expression. $\newline$ We can use the property that $\log(\frac{a}{b}) = \log(a) - \log(b)$ to rewrite the given logarithm as: $\newline$ $\log_{3}(1) - \log_{3}(45)$
Evaluating $\log(1)$ : Evaluate $\log_{3}(1)$ . $\newline$ The logarithm of $1$ to any base is $0$ because any number to the power of $0$ is $1$ . So, $\log_{3}(1) = 0$ .
Factoring $45$ : Factor $45$ to express it as a product of powers of $3$ . $\newline$ $45$ can be factored into $3 \times 15$ , and $15$ can be further factored into $3 \times 5$ . So, $45 = 3^2 \times 5$ .
Separating the Factors: Use the logarithm properties to separate the factors. $\newline$ We can use the property that $\log(ab) = \log(a) + \log(b)$ to rewrite $\log_{3}(45)$ as: $\newline$ $\log_{3}(3^2) + \log_{3}(5)$
Evaluating $\log(3^2)$ : Evaluate $\log_{3}(3^2)$ . $\newline$ Since the base of the logarithm and the base of the exponent are the same, $\log_{3}(3^2) = 2$ .
Evaluating $\log(5)$ : Evaluate $\log_{3}(5)$ . $\newline$ Since $5$ is not a power of $3$ , we cannot simplify this logarithm further without a calculator. We will need to use a calculator to find the value of $\log_{3}(5)$ .
Calculating $\log(3^2)$ : Calculate $\log_{3}(5)$ using a calculator. $\newline$ Using a calculator, we find that $\log_{3}(5) \approx 1.465$ (rounded to the nearest thousandth).
Combining the Results: Combine the results to find the final value of the original logarithm. $\newline$ Now we combine the results from steps $3$ , $6$ , and $8$ : $\newline$ $\log_{3}\left(\frac{1}{45}\right) = \log_{3}(1) - (\log_{3}(3^2) + \log_{3}(5))$ $\newline$ $= 0 - (2 + 1.465)$ $\newline$ $= -2 - 1.465$ $\newline$ $= -3.465$
Rounding the Final Answer: Round the final answer to the nearest thousandth. $\newline$ The final answer is already rounded to the nearest thousandth, so the final answer is $-3.465$ .